Statement

Consider the finite and infinite sequences of binary digits. Given an infinite sequence α\alpha and a natural numbernn, let α¯n\bar \alpha n be the finite sequence consisting of the first nn elements of α\alpha.

Let BB be a collection of finite sequences of bits, that is a subset of the free monoid on the boolean domain. Given an infinite sequence α\alpha and a natural number nn, we say that α\alphann-barsBB if α¯n∈B\bar \alpha n \in B; given only α\alpha, we say that α\alphabarsBB if α\alphann-bars BB for some nn.

We are interested in these three properties of BB:

BB is decidable: For every finite sequence uu, either u∈Bu \in B or u∉Bu \notin B. (This is trivial in classical logic but may hold constructively for a particular subset BB.)

BB is barred: For every infinite sequence α\alpha, α\alpha bars BB.

BB is uniform: For some natural number MM, for every infinite sequence α\alpha, if α\alpha bars BB at all, then α\alphann-bars BB for some n≤Mn \leq M.

A bar is a barred subset BB.

Fan Theorem

Every decidable bar is uniform.

Although the fan theorem is about bars, it is different from the bar theorem, which is related but stronger.

Obfuscation

Let 𝔹\mathbb{B} be the set {0,1}\{0,1\} of binary digits (bits) and ℕ\mathbb{N} the set {0,1,2,…}\{0,1,2,\ldots\} of natural numbers (numbers). Given a setAA, let A*A^* be the set of finite sequences of elements of AA, let AℕA^{\mathbb{N}} be the set of infinite sequences of elements of AA, and let 𝒫ΔA\mathcal{P}_{\Delta}A be the set of decidable subsets of AA. Then the fan theorem is about (elements of) 𝔹*\mathbb{B}^*, 𝔹ℕ\mathbb{B}^{\mathbb{N}}, and 𝒫Δ𝔹*\mathcal{P}_{\Delta}\mathbb{B}^*.

However, the sets ℕ\mathbb{N}, 𝔹*\mathbb{B}^*, and ℕ*\mathbb{N}^* are all isomorphic. Similarly, the sets 𝔹ℕ\mathbb{B}^{\mathbb{N}}, 𝒫Δℕ\mathcal{P}_{\Delta}\mathbb{N}, 𝒫Δ𝔹*\mathcal{P}_{\Delta}\mathbb{B}^*, and 𝒫Δℕ*\mathcal{P}_{\Delta}\mathbb{N}^* are all isomorphic. In much of the literature on bars, one tacitly uses all of these isomorphisms, taking ℕ\mathbb{N} and 𝔹ℕ\mathbb{B}^{\mathbb{N}} as chosen representatives of their isomorphism classes. Thus, everything in sight is either a natural number or an infinite sequence of bits.

The fan theorem is hard enough to understand when α\alpha is an infinite sequence of bits and α¯n\bar \alpha n is a finite sequence of bits; it is even harder to understand when α¯n\bar \alpha n is a natural number that bears no immediate relationship to the digits in the sequence α\alpha.

Variations

The fan theorem may be stated about all bars, not just the decidable ones. Brouwer himself at one point claimed that it held for all bars, but later Kleene showed that this contradicted Brouwer's continuity theorem?. However, the theorem does hold for all bars classically.

Use in analysis

In classical mathematics, the fan theorem is simply true.

In constructive mathematics, the fan theorem is equivalent to any and all of the following statements:

Some of the results above may use countable choice, but probably no more than AC0,0AC_{0,0} (which is choice for relations between ℕ\mathbb{N} and itself).

Uselessness in analysis

Point-wise real analysis without the fan theorem is very difficult, as the example from Waaldijk shows. This was Brouwer's motivation for introducing the fan theorem. But if you use locales (or other pointless approaches), then you don't need the fan theorem (or bar theorem).